rotating linked torri with arrow mapping
That can be something like this (if I understand the question...):

The idea is to hook a grid (1x2 proportions) then simple deform bend twice to make a torus and finally weld to remove doubles.
Then if we move the hook object (only in X/Y axis), that will rotate around the simple deform with bend option.

In short:

Hook will move all vertices (in the animation, just move the hook empty in X/Y so that the arrow will move around the torus)
Simple deform / bend 360 makes a tube
Simple deform / bend 360 makes a torus
Weld merges doubles